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PAPER-I: CLASSICAL MECHANICS & MATHEMATICAL PHYSICS

Unit I
: Canonical Transformation Equations of Canonical Transformations-Lagrange and Poisson’s Brackets-Invariance- Equation of Motion in Poisson Bracket Notation. Hamilton-Jacobi Theory: H-J Equation for Hamilton’s Principle Function-Hamilton’s Characteristic Equation- Separation of Variables-Harmonic Oscillator Problem in H-J Method-Action Angle Variables- Kepler Problem in Action Angle Variables.

Unit II
: Rigid Body Dynamics Generalized coordinates for Rigid Body Motion-Euler Angles-Angular Velocity, Angular Momentum of a rigid body-Moments and Products of Inertia-Principal Axes Transformation- Rotational Kinetic Energy-Moment of Inertia of a Rigid Body-Equation of Motion of a Rigid Body-Euler’s Equations

Unit III
: Mechanics of Small Oscillations Stable and Unstable Equilibrium-Two Coupled Oscillators-Formulation of the Problem- Properties of T,V and ω-Normal Coordinates and Normal Frequencies of Vibration-Systems with few Degrees of Freedom-Parallel Pendula-Double Pendulum-Triple Pendulum(degenerate system)-Linear Triatomic Molecule.

Unit IV
: Special Functions Legendre’s Polynomials and Functions- Differential Equations and Solutions-Rodriguez Formula-Generating Functions-Orthogonality-Relation between Legendre Polynomial and their Derivatives-Recurrence Relations-Bessel’s Function-Differential Equation and Solution- Generating Functions-Recurrence Relations Complex Variable Theory: Functions of a Complex Variable-Single and Multivalued Functions-Cauchy-Reimann Differential Equation-Analytical Line Integrals of Complex Function-Cauchy’s Integral Theorem and Integral Formula-Derivatives of an Analytic Function-Taylor’s Variables-Residue and Cauchy’s Residue Theorem-Application to the Equation of Definite Integrals-Conformal Transformations-Invariance of the Laplacian.

Unit V
: Linear Space Definition of Vector Space-Linear Dependence-Linear Independence-Basis-Dimension of a Vector Space-Representation of Vectors and Linear Operators with respect to Basis-Schmidt Orthogonalization Process-Inner Product. Fourier Series & Laplace Transforms: Fourier Series-Dirichlet’s Theorem-Change of Interval-Complex Form-Fourier Series in the Interval (0, θ) - Uses of Fourier Series.-Laplace Transform-Definition-Properties-Translation Property-Inverse Laplace Transform-Properties

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